Question: Perform the row operation, $9R_3\rightarrow R_3$, on the following matrix. $\left[\begin{array} {ccc} -5 & -7 & 8 & 5 \\ 1 & 2 & -3 & 2 \\ 1 & 0 & 0 & -1 \end{array} \right] $
Solution: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_3$ is given below. $R_3=\left[\begin{array} {ccc} 1 & 0 & 0 & -1 \end{array} \right]$ We are asked to perform the row operation, $9R_3\rightarrow R_3$. Therefore, we must multiply $R_3$ by $9$. $\begin{aligned}9R_3 &= 9\left[\begin{array} {ccc} 1 & 0 & 0 & -1 \end{array} \right] \\\\&=\left[\begin{array} {ccc} 9 & 0 & 0 & -9 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_3$ with $9R_3$. $\left[\begin{array} {ccc} -5 & -7 & 8 & 5 \\ 1 & 2 & -3 & 2 \\ {1} & {0} & {0} & {-1} \end{array} \right] \xrightarrow{9R_3\rightarrow R_3} \left[\begin{array} {ccc} -5 & -7 & 8 & 5 \\ 1 & 2 & -3 & 2 \\ {9} & {0} & {0} & {-9} \end{array} \right] $ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} -5 & -7 & 8 & 5 \\ 1 & 2 & -3 & 2 \\ 9 & 0 & 0 & -9 \end{array} \right]$